If a Blazing Torch controlled by one player somehow winds up equipping a creature a different player controls, the damage ability can’t be activated by either player. Only the creature’s controller may activate the ability — but since that player can’t sacrifice Blazing Torch (a permanent they don’t control), the ability’s cost can’t be paid.

The source of the damage is Blazing Torch, not the equipped creature. However, the equipped creature’s ability is what targets the permanent or player. If Blazing Torch is equipped to a red creature, for example, the ability couldn’t target a creature with protection from red. It could target a creature with protection from artifacts, but all the damage would be prevented.